In control theory and in particular when studying the properties of a linear time-invariant system in state space form, the Hautus lemma (after Malo L. J. Hautus), also commonly known as the Popov-Belevitch-Hautus test or PBH test,[1][2] can prove to be a powerful tool.

A special case of this result appeared first in 1963 in a paper by Elmer G. Gilbert,[1] and was later expanded to the current PHB test with contributions by Vasile M. Popov in 1966,[3][4] Vitold Belevitch in 1968,[5] and Malo Hautus in 1969,[5] who emphasized its applicability in proving results for linear time-invariant systems.

Statement

There exist multiple forms of the lemma:

Hautus Lemma for controllability

The Hautus lemma for controllability says that given a square matrix and a the following are equivalent:

  1. The pair is controllable
  2. For all it holds that
  3. For all that are eigenvalues of it holds that

Hautus Lemma for stabilizability

The Hautus lemma for stabilizability says that given a square matrix and a the following are equivalent:

  1. The pair is stabilizable
  2. For all that are eigenvalues of and for which it holds that

Hautus Lemma for observability

The Hautus lemma for observability says that given a square matrix and a the following are equivalent:

  1. The pair is observable.
  2. For all it holds that
  3. For all that are eigenvalues of it holds that

Hautus Lemma for detectability

The Hautus lemma for detectability says that given a square matrix and a the following are equivalent:

  1. The pair is detectable
  2. For all that are eigenvalues of and for which it holds that

References

  • Sontag, Eduard D. (1998). Mathematical Control Theory: Deterministic Finite-Dimensional Systems. New York: Springer. ISBN 0-387-98489-5.
  • Zabczyk, Jerzy (1995). Mathematical Control Theory – An Introduction. Boston: Birkhauser. ISBN 3-7643-3645-5.

Notes

  1. 1 2 Hespanha, Joao (2018). Linear Systems Theory (Second ed.). Princeton University Press. ISBN 9780691179575.
  2. ↑ Bernstein, Dennis S. (2018). Scalar, Vector, and Matrix Mathematics: Theory, Facts, and Formulas (Revised and expanded ed.). Princeton University Press. ISBN 9780691151205.
  3. ↑ Popov, Vasile Mihai (1966). Hiperstabilitatea sistemelor automate [Hyperstability of Control Systems]. Editura Academiei Republicii Socialiste România.
  4. ↑ Popov, V.M. (1973). Hyperstability of Control Systems. Berlin: Springer-Verlag.
  5. 1 2 Belevitch, V. (1968). Classical Network Theory. San Francisco: Holden–Day.
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